Memristive devices, or memristors, are a new type of switchable devices with an electrically switchable device resistance. Memristive devices are both scientifically and technically interesting, and hold promise for non-volatile memory (NVM) and other fields. For NVM applications, many memristive devices can be formed in a two-dimensional array, such as a crossbar structure, to provide a very high storage capacity. Nevertheless, it has been a major challenge to switch a selected memristive device within a crossbar structure reliably to desired states, due to the large leakage current through other memristive devices in the array that are connected to the same row or column line of the device being switched.